19 - Worked Examples and Concept Example Usage in Understanding Mathematical Concepts and Proofs

Summary

Elsewhere in this volume, Watson and Mason discuss example generation from the students' perspective by highlighting some of the ways that example generation can be used to increase students' understanding of mathematics and improve their attitudes toward mathematics. This chapter complements this work by describing ways that teachers and textbooks might use examples to help undergraduates better understand mathematics. We distinguish between using worked examples to solve exercises and problems and using examples to help promote students' understanding of mathematical concepts and proofs. We begin with worked examples provided by the teacher or textbook. We then discuss the role of examples in building an understanding of a mathematical concept. Next we discuss how examples can be useful in understanding mathematical proofs. In each of these sections, we present specific suggestions that teachers might use in their own mathematics classrooms and we cite research studies that motivate and support these suggestions.

Worked Examples

The term “example” has multiple uses in mathematics education (cf., Watson & Mason, 2002). In some contexts, the word “example” refers to an illustration of a technique used to complete a certain type of mathematical task. For instance, a written solution to the question “Find all local minima and maxima of the function f(x)=x³+5x²-8” might be regarded as an example of how to solve minimum/maximum problems in an introductory calculus course. This is the way that the word example is often used in undergraduate textbooks, in which individual sections of the book frequently introduce a technique and then provide a series of examples in which the technique is applied.